\[ \int \frac {1}{\left (a+b x^4\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {x}{6 a \left (b \,x^{4}+a \right )^{\frac {3}{2}}}+\frac {5 x}{12 a^{2} \sqrt {b \,x^{4}+a}}+\frac {5 \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x^{2} \sqrt {b}\right ) \sqrt {\frac {b \,x^{4}+a}{\left (\sqrt {a}+x^{2} \sqrt {b}\right )^{2}}}}{24 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right ) a^{\frac {9}{4}} b^{\frac {1}{4}} \sqrt {b \,x^{4}+a}} \]
command
integrate(1/(b*x^4+a)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {5 \, {\left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )} \sqrt {a} \left (-\frac {b}{a}\right )^{\frac {3}{4}} {\rm ellipticF}\left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}, -1\right ) - {\left (5 \, b^{2} x^{5} + 7 \, a b x\right )} \sqrt {b x^{4} + a}}{12 \, {\left (a^{2} b^{3} x^{8} + 2 \, a^{3} b^{2} x^{4} + a^{4} b\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {b x^{4} + a}}{b^{3} x^{12} + 3 \, a b^{2} x^{8} + 3 \, a^{2} b x^{4} + a^{3}}, x\right ) \]